Theorem ltexprlemloc 7722

Description: Our constructed difference is located. Lemma for ltexpri 7728. (Contributed by Jim Kingdon, 17-Dec-2019.)

Hypothesis

Ref	Expression
ltexprlem.1	|- C = <. { x e. Q. | E. y ( y e. ( 2nd ` A ) /\ ( y +Q x ) e. ( 1st ` B ) ) } , { x e. Q. | E. y ( y e. ( 1st ` A ) /\ ( y +Q x ) e. ( 2nd ` B ) ) } >.

Assertion

Ref	Expression
ltexprlemloc	|- ( A <P B -> A. q e. Q. A. r e. Q. ( q <Q r -> ( q e. ( 1st ` C ) \/ r e. ( 2nd ` C ) ) ) )

Distinct variable groups:   x, y, q, r, A   x, B, y, q, r   x, C, y, q, r

Proof of Theorem ltexprlemloc

Dummy variables z w f g h are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ltexnqi 7524	. . . . . 6 |- ( q <Q r -> E. w e. Q. ( q +Q w ) = r )
2	1	adantl 277	. . . . 5 |- ( ( A <P B /\ q <Q r ) -> E. w e. Q. ( q +Q w ) = r )
3		ltrelpr 7620	. . . . . . . . . 10 |- <P C_ ( P. X. P. )
4	3	brel 4728	. . . . . . . . 9 |- ( A <P B -> ( A e. P. /\ B e. P. ) )
5	4	simpld 112	. . . . . . . 8 |- ( A <P B -> A e. P. )
6		prop 7590	. . . . . . . . 9 |- ( A e. P. -> <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. )
7		prarloc 7618	. . . . . . . . 9 |- ( ( <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. /\ w e. Q. ) -> E. z e. ( 1st ` A ) E. y e. ( 2nd ` A ) y <Q ( z +Q w ) )
8	6, 7	sylan 283	. . . . . . . 8 |- ( ( A e. P. /\ w e. Q. ) -> E. z e. ( 1st ` A ) E. y e. ( 2nd ` A ) y <Q ( z +Q w ) )
9	5, 8	sylan 283	. . . . . . 7 |- ( ( A <P B /\ w e. Q. ) -> E. z e. ( 1st ` A ) E. y e. ( 2nd ` A ) y <Q ( z +Q w ) )
10	9	ad2ant2r 509	. . . . . 6 |- ( ( ( A <P B /\ q <Q r ) /\ ( w e. Q. /\ ( q +Q w ) = r ) ) -> E. z e. ( 1st ` A ) E. y e. ( 2nd ` A ) y <Q ( z +Q w ) )
11	4	simprd 114	. . . . . . . . . . . . . 14 |- ( A <P B -> B e. P. )
12	11	ad2antrr 488	. . . . . . . . . . . . 13 |- ( ( ( A <P B /\ q <Q r ) /\ ( w e. Q. /\ ( q +Q w ) = r ) ) -> B e. P. )
13	12	ad2antrr 488	. . . . . . . . . . . 12 |- ( ( ( ( ( A <P B /\ q <Q r ) /\ ( w e. Q. /\ ( q +Q w ) = r ) ) /\ ( z e. ( 1st ` A ) /\ y e. ( 2nd ` A ) ) ) /\ y <Q ( z +Q w ) ) -> B e. P. )
14		ltanqg 7515	. . . . . . . . . . . . . . . 16 |- ( ( f e. Q. /\ g e. Q. /\ h e. Q. ) -> ( f <Q g <-> ( h +Q f ) <Q ( h +Q g ) ) )
15	14	adantl 277	. . . . . . . . . . . . . . 15 |- ( ( ( ( ( A <P B /\ q <Q r ) /\ ( w e. Q. /\ ( q +Q w ) = r ) ) /\ ( z e. ( 1st ` A ) /\ y e. ( 2nd ` A ) ) ) /\ ( f e. Q. /\ g e. Q. /\ h e. Q. ) ) -> ( f <Q g <-> ( h +Q f ) <Q ( h +Q g ) ) )
16		elprnqu 7597	. . . . . . . . . . . . . . . . . . 19 |- ( ( <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. /\ y e. ( 2nd ` A ) ) -> y e. Q. )
17	6, 16	sylan 283	. . . . . . . . . . . . . . . . . 18 |- ( ( A e. P. /\ y e. ( 2nd ` A ) ) -> y e. Q. )
18	5, 17	sylan 283	. . . . . . . . . . . . . . . . 17 |- ( ( A <P B /\ y e. ( 2nd ` A ) ) -> y e. Q. )
19	18	adantlr 477	. . . . . . . . . . . . . . . 16 |- ( ( ( A <P B /\ q <Q r ) /\ y e. ( 2nd ` A ) ) -> y e. Q. )
20	19	ad2ant2rl 511	. . . . . . . . . . . . . . 15 |- ( ( ( ( A <P B /\ q <Q r ) /\ ( w e. Q. /\ ( q +Q w ) = r ) ) /\ ( z e. ( 1st ` A ) /\ y e. ( 2nd ` A ) ) ) -> y e. Q. )
21		elprnql 7596	. . . . . . . . . . . . . . . . . . . 20 |- ( ( <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. /\ z e. ( 1st ` A ) ) -> z e. Q. )
22	6, 21	sylan 283	. . . . . . . . . . . . . . . . . . 19 |- ( ( A e. P. /\ z e. ( 1st ` A ) ) -> z e. Q. )
23	5, 22	sylan 283	. . . . . . . . . . . . . . . . . 18 |- ( ( A <P B /\ z e. ( 1st ` A ) ) -> z e. Q. )
24	23	adantlr 477	. . . . . . . . . . . . . . . . 17 |- ( ( ( A <P B /\ q <Q r ) /\ z e. ( 1st ` A ) ) -> z e. Q. )
25	24	ad2ant2r 509	. . . . . . . . . . . . . . . 16 |- ( ( ( ( A <P B /\ q <Q r ) /\ ( w e. Q. /\ ( q +Q w ) = r ) ) /\ ( z e. ( 1st ` A ) /\ y e. ( 2nd ` A ) ) ) -> z e. Q. )
26		simplrl 535	. . . . . . . . . . . . . . . 16 |- ( ( ( ( A <P B /\ q <Q r ) /\ ( w e. Q. /\ ( q +Q w ) = r ) ) /\ ( z e. ( 1st ` A ) /\ y e. ( 2nd ` A ) ) ) -> w e. Q. )
27		addclnq 7490	. . . . . . . . . . . . . . . 16 |- ( ( z e. Q. /\ w e. Q. ) -> ( z +Q w ) e. Q. )
28	25, 26, 27	syl2anc 411	. . . . . . . . . . . . . . 15 |- ( ( ( ( A <P B /\ q <Q r ) /\ ( w e. Q. /\ ( q +Q w ) = r ) ) /\ ( z e. ( 1st ` A ) /\ y e. ( 2nd ` A ) ) ) -> ( z +Q w ) e. Q. )
29		ltrelnq 7480	. . . . . . . . . . . . . . . . . . 19 |- <Q C_ ( Q. X. Q. )
30	29	brel 4728	. . . . . . . . . . . . . . . . . 18 |- ( q <Q r -> ( q e. Q. /\ r e. Q. ) )
31	30	simpld 112	. . . . . . . . . . . . . . . . 17 |- ( q <Q r -> q e. Q. )
32	31	adantl 277	. . . . . . . . . . . . . . . 16 |- ( ( A <P B /\ q <Q r ) -> q e. Q. )
33	32	ad2antrr 488	. . . . . . . . . . . . . . 15 |- ( ( ( ( A <P B /\ q <Q r ) /\ ( w e. Q. /\ ( q +Q w ) = r ) ) /\ ( z e. ( 1st ` A ) /\ y e. ( 2nd ` A ) ) ) -> q e. Q. )
34		addcomnqg 7496	. . . . . . . . . . . . . . . 16 |- ( ( f e. Q. /\ g e. Q. ) -> ( f +Q g ) = ( g +Q f ) )
35	34	adantl 277	. . . . . . . . . . . . . . 15 |- ( ( ( ( ( A <P B /\ q <Q r ) /\ ( w e. Q. /\ ( q +Q w ) = r ) ) /\ ( z e. ( 1st ` A ) /\ y e. ( 2nd ` A ) ) ) /\ ( f e. Q. /\ g e. Q. ) ) -> ( f +Q g ) = ( g +Q f ) )
36	15, 20, 28, 33, 35	caovord2d 6118	. . . . . . . . . . . . . 14 |- ( ( ( ( A <P B /\ q <Q r ) /\ ( w e. Q. /\ ( q +Q w ) = r ) ) /\ ( z e. ( 1st ` A ) /\ y e. ( 2nd ` A ) ) ) -> ( y <Q ( z +Q w ) <-> ( y +Q q ) <Q ( ( z +Q w ) +Q q ) ) )
37		addassnqg 7497	. . . . . . . . . . . . . . . . 17 |- ( ( z e. Q. /\ w e. Q. /\ q e. Q. ) -> ( ( z +Q w ) +Q q ) = ( z +Q ( w +Q q ) ) )
38	25, 26, 33, 37	syl3anc 1250	. . . . . . . . . . . . . . . 16 |- ( ( ( ( A <P B /\ q <Q r ) /\ ( w e. Q. /\ ( q +Q w ) = r ) ) /\ ( z e. ( 1st ` A ) /\ y e. ( 2nd ` A ) ) ) -> ( ( z +Q w ) +Q q ) = ( z +Q ( w +Q q ) ) )
39		addcomnqg 7496	. . . . . . . . . . . . . . . . . 18 |- ( ( w e. Q. /\ q e. Q. ) -> ( w +Q q ) = ( q +Q w ) )
40	26, 33, 39	syl2anc 411	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( A <P B /\ q <Q r ) /\ ( w e. Q. /\ ( q +Q w ) = r ) ) /\ ( z e. ( 1st ` A ) /\ y e. ( 2nd ` A ) ) ) -> ( w +Q q ) = ( q +Q w ) )
41	40	oveq2d 5962	. . . . . . . . . . . . . . . 16 |- ( ( ( ( A <P B /\ q <Q r ) /\ ( w e. Q. /\ ( q +Q w ) = r ) ) /\ ( z e. ( 1st ` A ) /\ y e. ( 2nd ` A ) ) ) -> ( z +Q ( w +Q q ) ) = ( z +Q ( q +Q w ) ) )
42		simplrr 536	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( A <P B /\ q <Q r ) /\ ( w e. Q. /\ ( q +Q w ) = r ) ) /\ ( z e. ( 1st ` A ) /\ y e. ( 2nd ` A ) ) ) -> ( q +Q w ) = r )
43	42	oveq2d 5962	. . . . . . . . . . . . . . . 16 |- ( ( ( ( A <P B /\ q <Q r ) /\ ( w e. Q. /\ ( q +Q w ) = r ) ) /\ ( z e. ( 1st ` A ) /\ y e. ( 2nd ` A ) ) ) -> ( z +Q ( q +Q w ) ) = ( z +Q r ) )
44	38, 41, 43	3eqtrd 2242	. . . . . . . . . . . . . . 15 |- ( ( ( ( A <P B /\ q <Q r ) /\ ( w e. Q. /\ ( q +Q w ) = r ) ) /\ ( z e. ( 1st ` A ) /\ y e. ( 2nd ` A ) ) ) -> ( ( z +Q w ) +Q q ) = ( z +Q r ) )
45	44	breq2d 4057	. . . . . . . . . . . . . 14 |- ( ( ( ( A <P B /\ q <Q r ) /\ ( w e. Q. /\ ( q +Q w ) = r ) ) /\ ( z e. ( 1st ` A ) /\ y e. ( 2nd ` A ) ) ) -> ( ( y +Q q ) <Q ( ( z +Q w ) +Q q ) <-> ( y +Q q ) <Q ( z +Q r ) ) )
46	36, 45	bitrd 188	. . . . . . . . . . . . 13 |- ( ( ( ( A <P B /\ q <Q r ) /\ ( w e. Q. /\ ( q +Q w ) = r ) ) /\ ( z e. ( 1st ` A ) /\ y e. ( 2nd ` A ) ) ) -> ( y <Q ( z +Q w ) <-> ( y +Q q ) <Q ( z +Q r ) ) )
47	46	biimpa 296	. . . . . . . . . . . 12 |- ( ( ( ( ( A <P B /\ q <Q r ) /\ ( w e. Q. /\ ( q +Q w ) = r ) ) /\ ( z e. ( 1st ` A ) /\ y e. ( 2nd ` A ) ) ) /\ y <Q ( z +Q w ) ) -> ( y +Q q ) <Q ( z +Q r ) )
48		prop 7590	. . . . . . . . . . . . 13 |- ( B e. P. -> <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. )
49		prloc 7606	. . . . . . . . . . . . 13 |- ( ( <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. /\ ( y +Q q ) <Q ( z +Q r ) ) -> ( ( y +Q q ) e. ( 1st ` B ) \/ ( z +Q r ) e. ( 2nd ` B ) ) )
50	48, 49	sylan 283	. . . . . . . . . . . 12 |- ( ( B e. P. /\ ( y +Q q ) <Q ( z +Q r ) ) -> ( ( y +Q q ) e. ( 1st ` B ) \/ ( z +Q r ) e. ( 2nd ` B ) ) )
51	13, 47, 50	syl2anc 411	. . . . . . . . . . 11 |- ( ( ( ( ( A <P B /\ q <Q r ) /\ ( w e. Q. /\ ( q +Q w ) = r ) ) /\ ( z e. ( 1st ` A ) /\ y e. ( 2nd ` A ) ) ) /\ y <Q ( z +Q w ) ) -> ( ( y +Q q ) e. ( 1st ` B ) \/ ( z +Q r ) e. ( 2nd ` B ) ) )
52	51	ex 115	. . . . . . . . . 10 |- ( ( ( ( A <P B /\ q <Q r ) /\ ( w e. Q. /\ ( q +Q w ) = r ) ) /\ ( z e. ( 1st ` A ) /\ y e. ( 2nd ` A ) ) ) -> ( y <Q ( z +Q w ) -> ( ( y +Q q ) e. ( 1st ` B ) \/ ( z +Q r ) e. ( 2nd ` B ) ) ) )
53	52	anassrs 400	. . . . . . . . 9 |- ( ( ( ( ( A <P B /\ q <Q r ) /\ ( w e. Q. /\ ( q +Q w ) = r ) ) /\ z e. ( 1st ` A ) ) /\ y e. ( 2nd ` A ) ) -> ( y <Q ( z +Q w ) -> ( ( y +Q q ) e. ( 1st ` B ) \/ ( z +Q r ) e. ( 2nd ` B ) ) ) )
54	53	reximdva 2608	. . . . . . . 8 |- ( ( ( ( A <P B /\ q <Q r ) /\ ( w e. Q. /\ ( q +Q w ) = r ) ) /\ z e. ( 1st ` A ) ) -> ( E. y e. ( 2nd ` A ) y <Q ( z +Q w ) -> E. y e. ( 2nd ` A ) ( ( y +Q q ) e. ( 1st ` B ) \/ ( z +Q r ) e. ( 2nd ` B ) ) ) )
55	54	reximdva 2608	. . . . . . 7 |- ( ( ( A <P B /\ q <Q r ) /\ ( w e. Q. /\ ( q +Q w ) = r ) ) -> ( E. z e. ( 1st ` A ) E. y e. ( 2nd ` A ) y <Q ( z +Q w ) -> E. z e. ( 1st ` A ) E. y e. ( 2nd ` A ) ( ( y +Q q ) e. ( 1st ` B ) \/ ( z +Q r ) e. ( 2nd ` B ) ) ) )
56		prml 7592	. . . . . . . . . . . 12 |- ( <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. -> E. z e. Q. z e. ( 1st ` A ) )
57		rexex 2552	. . . . . . . . . . . 12 |- ( E. z e. Q. z e. ( 1st ` A ) -> E. z z e. ( 1st ` A ) )
58	6, 56, 57	3syl 17	. . . . . . . . . . 11 |- ( A e. P. -> E. z z e. ( 1st ` A ) )
59		r19.45mv 3554	. . . . . . . . . . 11 |- ( E. z z e. ( 1st ` A ) -> ( E. z e. ( 1st ` A ) ( E. y e. ( 2nd ` A ) ( y +Q q ) e. ( 1st ` B ) \/ ( z +Q r ) e. ( 2nd ` B ) ) <-> ( E. y e. ( 2nd ` A ) ( y +Q q ) e. ( 1st ` B ) \/ E. z e. ( 1st ` A ) ( z +Q r ) e. ( 2nd ` B ) ) ) )
60	5, 58, 59	3syl 17	. . . . . . . . . 10 |- ( A <P B -> ( E. z e. ( 1st ` A ) ( E. y e. ( 2nd ` A ) ( y +Q q ) e. ( 1st ` B ) \/ ( z +Q r ) e. ( 2nd ` B ) ) <-> ( E. y e. ( 2nd ` A ) ( y +Q q ) e. ( 1st ` B ) \/ E. z e. ( 1st ` A ) ( z +Q r ) e. ( 2nd ` B ) ) ) )
61	60	adantr 276	. . . . . . . . 9 |- ( ( A <P B /\ q <Q r ) -> ( E. z e. ( 1st ` A ) ( E. y e. ( 2nd ` A ) ( y +Q q ) e. ( 1st ` B ) \/ ( z +Q r ) e. ( 2nd ` B ) ) <-> ( E. y e. ( 2nd ` A ) ( y +Q q ) e. ( 1st ` B ) \/ E. z e. ( 1st ` A ) ( z +Q r ) e. ( 2nd ` B ) ) ) )
62		prmu 7593	. . . . . . . . . . . . 13 |- ( <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. -> E. x e. Q. x e. ( 2nd ` A ) )
63		rexex 2552	. . . . . . . . . . . . 13 |- ( E. x e. Q. x e. ( 2nd ` A ) -> E. x x e. ( 2nd ` A ) )
64	6, 62, 63	3syl 17	. . . . . . . . . . . 12 |- ( A e. P. -> E. x x e. ( 2nd ` A ) )
65		r19.43 2664	. . . . . . . . . . . . 13 |- ( E. y e. ( 2nd ` A ) ( ( y +Q q ) e. ( 1st ` B ) \/ ( z +Q r ) e. ( 2nd ` B ) ) <-> ( E. y e. ( 2nd ` A ) ( y +Q q ) e. ( 1st ` B ) \/ E. y e. ( 2nd ` A ) ( z +Q r ) e. ( 2nd ` B ) ) )
66		r19.9rmv 3552	. . . . . . . . . . . . . 14 |- ( E. x x e. ( 2nd ` A ) -> ( ( z +Q r ) e. ( 2nd ` B ) <-> E. y e. ( 2nd ` A ) ( z +Q r ) e. ( 2nd ` B ) ) )
67	66	orbi2d 792	. . . . . . . . . . . . 13 |- ( E. x x e. ( 2nd ` A ) -> ( ( E. y e. ( 2nd ` A ) ( y +Q q ) e. ( 1st ` B ) \/ ( z +Q r ) e. ( 2nd ` B ) ) <-> ( E. y e. ( 2nd ` A ) ( y +Q q ) e. ( 1st ` B ) \/ E. y e. ( 2nd ` A ) ( z +Q r ) e. ( 2nd ` B ) ) ) )
68	65, 67	bitr4id 199	. . . . . . . . . . . 12 |- ( E. x x e. ( 2nd ` A ) -> ( E. y e. ( 2nd ` A ) ( ( y +Q q ) e. ( 1st ` B ) \/ ( z +Q r ) e. ( 2nd ` B ) ) <-> ( E. y e. ( 2nd ` A ) ( y +Q q ) e. ( 1st ` B ) \/ ( z +Q r ) e. ( 2nd ` B ) ) ) )
69	5, 64, 68	3syl 17	. . . . . . . . . . 11 |- ( A <P B -> ( E. y e. ( 2nd ` A ) ( ( y +Q q ) e. ( 1st ` B ) \/ ( z +Q r ) e. ( 2nd ` B ) ) <-> ( E. y e. ( 2nd ` A ) ( y +Q q ) e. ( 1st ` B ) \/ ( z +Q r ) e. ( 2nd ` B ) ) ) )
70	69	rexbidv 2507	. . . . . . . . . 10 |- ( A <P B -> ( E. z e. ( 1st ` A ) E. y e. ( 2nd ` A ) ( ( y +Q q ) e. ( 1st ` B ) \/ ( z +Q r ) e. ( 2nd ` B ) ) <-> E. z e. ( 1st ` A ) ( E. y e. ( 2nd ` A ) ( y +Q q ) e. ( 1st ` B ) \/ ( z +Q r ) e. ( 2nd ` B ) ) ) )
71	70	adantr 276	. . . . . . . . 9 |- ( ( A <P B /\ q <Q r ) -> ( E. z e. ( 1st ` A ) E. y e. ( 2nd ` A ) ( ( y +Q q ) e. ( 1st ` B ) \/ ( z +Q r ) e. ( 2nd ` B ) ) <-> E. z e. ( 1st ` A ) ( E. y e. ( 2nd ` A ) ( y +Q q ) e. ( 1st ` B ) \/ ( z +Q r ) e. ( 2nd ` B ) ) ) )
72		ltexprlem.1	. . . . . . . . . . . . . 14 |- C = <. { x e. Q. | E. y ( y e. ( 2nd ` A ) /\ ( y +Q x ) e. ( 1st ` B ) ) } , { x e. Q. | E. y ( y e. ( 1st ` A ) /\ ( y +Q x ) e. ( 2nd ` B ) ) } >.
73	72	ltexprlemell 7713	. . . . . . . . . . . . 13 |- ( q e. ( 1st ` C ) <-> ( q e. Q. /\ E. y ( y e. ( 2nd ` A ) /\ ( y +Q q ) e. ( 1st ` B ) ) ) )
74	72	ltexprlemelu 7714	. . . . . . . . . . . . . 14 |- ( r e. ( 2nd ` C ) <-> ( r e. Q. /\ E. y ( y e. ( 1st ` A ) /\ ( y +Q r ) e. ( 2nd ` B ) ) ) )
75		eleq1 2268	. . . . . . . . . . . . . . . . 17 |- ( y = z -> ( y e. ( 1st ` A ) <-> z e. ( 1st ` A ) ) )
76		oveq1 5953	. . . . . . . . . . . . . . . . . 18 |- ( y = z -> ( y +Q r ) = ( z +Q r ) )
77	76	eleq1d 2274	. . . . . . . . . . . . . . . . 17 |- ( y = z -> ( ( y +Q r ) e. ( 2nd ` B ) <-> ( z +Q r ) e. ( 2nd ` B ) ) )
78	75, 77	anbi12d 473	. . . . . . . . . . . . . . . 16 |- ( y = z -> ( ( y e. ( 1st ` A ) /\ ( y +Q r ) e. ( 2nd ` B ) ) <-> ( z e. ( 1st ` A ) /\ ( z +Q r ) e. ( 2nd ` B ) ) ) )
79	78	cbvexv 1942	. . . . . . . . . . . . . . 15 |- ( E. y ( y e. ( 1st ` A ) /\ ( y +Q r ) e. ( 2nd ` B ) ) <-> E. z ( z e. ( 1st ` A ) /\ ( z +Q r ) e. ( 2nd ` B ) ) )
80	79	anbi2i 457	. . . . . . . . . . . . . 14 |- ( ( r e. Q. /\ E. y ( y e. ( 1st ` A ) /\ ( y +Q r ) e. ( 2nd ` B ) ) ) <-> ( r e. Q. /\ E. z ( z e. ( 1st ` A ) /\ ( z +Q r ) e. ( 2nd ` B ) ) ) )
81	74, 80	bitri 184	. . . . . . . . . . . . 13 |- ( r e. ( 2nd ` C ) <-> ( r e. Q. /\ E. z ( z e. ( 1st ` A ) /\ ( z +Q r ) e. ( 2nd ` B ) ) ) )
82	73, 81	orbi12i 766	. . . . . . . . . . . 12 |- ( ( q e. ( 1st ` C ) \/ r e. ( 2nd ` C ) ) <-> ( ( q e. Q. /\ E. y ( y e. ( 2nd ` A ) /\ ( y +Q q ) e. ( 1st ` B ) ) ) \/ ( r e. Q. /\ E. z ( z e. ( 1st ` A ) /\ ( z +Q r ) e. ( 2nd ` B ) ) ) ) )
83		ibar 301	. . . . . . . . . . . . . . 15 |- ( q e. Q. -> ( E. y ( y e. ( 2nd ` A ) /\ ( y +Q q ) e. ( 1st ` B ) ) <-> ( q e. Q. /\ E. y ( y e. ( 2nd ` A ) /\ ( y +Q q ) e. ( 1st ` B ) ) ) ) )
84	83	adantr 276	. . . . . . . . . . . . . 14 |- ( ( q e. Q. /\ r e. Q. ) -> ( E. y ( y e. ( 2nd ` A ) /\ ( y +Q q ) e. ( 1st ` B ) ) <-> ( q e. Q. /\ E. y ( y e. ( 2nd ` A ) /\ ( y +Q q ) e. ( 1st ` B ) ) ) ) )
85		ibar 301	. . . . . . . . . . . . . . 15 |- ( r e. Q. -> ( E. z ( z e. ( 1st ` A ) /\ ( z +Q r ) e. ( 2nd ` B ) ) <-> ( r e. Q. /\ E. z ( z e. ( 1st ` A ) /\ ( z +Q r ) e. ( 2nd ` B ) ) ) ) )
86	85	adantl 277	. . . . . . . . . . . . . 14 |- ( ( q e. Q. /\ r e. Q. ) -> ( E. z ( z e. ( 1st ` A ) /\ ( z +Q r ) e. ( 2nd ` B ) ) <-> ( r e. Q. /\ E. z ( z e. ( 1st ` A ) /\ ( z +Q r ) e. ( 2nd ` B ) ) ) ) )
87	84, 86	orbi12d 795	. . . . . . . . . . . . 13 |- ( ( q e. Q. /\ r e. Q. ) -> ( ( E. y ( y e. ( 2nd ` A ) /\ ( y +Q q ) e. ( 1st ` B ) ) \/ E. z ( z e. ( 1st ` A ) /\ ( z +Q r ) e. ( 2nd ` B ) ) ) <-> ( ( q e. Q. /\ E. y ( y e. ( 2nd ` A ) /\ ( y +Q q ) e. ( 1st ` B ) ) ) \/ ( r e. Q. /\ E. z ( z e. ( 1st ` A ) /\ ( z +Q r ) e. ( 2nd ` B ) ) ) ) ) )
88	30, 87	syl 14	. . . . . . . . . . . 12 |- ( q <Q r -> ( ( E. y ( y e. ( 2nd ` A ) /\ ( y +Q q ) e. ( 1st ` B ) ) \/ E. z ( z e. ( 1st ` A ) /\ ( z +Q r ) e. ( 2nd ` B ) ) ) <-> ( ( q e. Q. /\ E. y ( y e. ( 2nd ` A ) /\ ( y +Q q ) e. ( 1st ` B ) ) ) \/ ( r e. Q. /\ E. z ( z e. ( 1st ` A ) /\ ( z +Q r ) e. ( 2nd ` B ) ) ) ) ) )
89	82, 88	bitr4id 199	. . . . . . . . . . 11 |- ( q <Q r -> ( ( q e. ( 1st ` C ) \/ r e. ( 2nd ` C ) ) <-> ( E. y ( y e. ( 2nd ` A ) /\ ( y +Q q ) e. ( 1st ` B ) ) \/ E. z ( z e. ( 1st ` A ) /\ ( z +Q r ) e. ( 2nd ` B ) ) ) ) )
90		df-rex 2490	. . . . . . . . . . . 12 |- ( E. y e. ( 2nd ` A ) ( y +Q q ) e. ( 1st ` B ) <-> E. y ( y e. ( 2nd ` A ) /\ ( y +Q q ) e. ( 1st ` B ) ) )
91		df-rex 2490	. . . . . . . . . . . 12 |- ( E. z e. ( 1st ` A ) ( z +Q r ) e. ( 2nd ` B ) <-> E. z ( z e. ( 1st ` A ) /\ ( z +Q r ) e. ( 2nd ` B ) ) )
92	90, 91	orbi12i 766	. . . . . . . . . . 11 |- ( ( E. y e. ( 2nd ` A ) ( y +Q q ) e. ( 1st ` B ) \/ E. z e. ( 1st ` A ) ( z +Q r ) e. ( 2nd ` B ) ) <-> ( E. y ( y e. ( 2nd ` A ) /\ ( y +Q q ) e. ( 1st ` B ) ) \/ E. z ( z e. ( 1st ` A ) /\ ( z +Q r ) e. ( 2nd ` B ) ) ) )
93	89, 92	bitr4di 198	. . . . . . . . . 10 |- ( q <Q r -> ( ( q e. ( 1st ` C ) \/ r e. ( 2nd ` C ) ) <-> ( E. y e. ( 2nd ` A ) ( y +Q q ) e. ( 1st ` B ) \/ E. z e. ( 1st ` A ) ( z +Q r ) e. ( 2nd ` B ) ) ) )
94	93	adantl 277	. . . . . . . . 9 |- ( ( A <P B /\ q <Q r ) -> ( ( q e. ( 1st ` C ) \/ r e. ( 2nd ` C ) ) <-> ( E. y e. ( 2nd ` A ) ( y +Q q ) e. ( 1st ` B ) \/ E. z e. ( 1st ` A ) ( z +Q r ) e. ( 2nd ` B ) ) ) )
95	61, 71, 94	3bitr4rd 221	. . . . . . . 8 |- ( ( A <P B /\ q <Q r ) -> ( ( q e. ( 1st ` C ) \/ r e. ( 2nd ` C ) ) <-> E. z e. ( 1st ` A ) E. y e. ( 2nd ` A ) ( ( y +Q q ) e. ( 1st ` B ) \/ ( z +Q r ) e. ( 2nd ` B ) ) ) )
96	95	adantr 276	. . . . . . 7 |- ( ( ( A <P B /\ q <Q r ) /\ ( w e. Q. /\ ( q +Q w ) = r ) ) -> ( ( q e. ( 1st ` C ) \/ r e. ( 2nd ` C ) ) <-> E. z e. ( 1st ` A ) E. y e. ( 2nd ` A ) ( ( y +Q q ) e. ( 1st ` B ) \/ ( z +Q r ) e. ( 2nd ` B ) ) ) )
97	55, 96	sylibrd 169	. . . . . 6 |- ( ( ( A <P B /\ q <Q r ) /\ ( w e. Q. /\ ( q +Q w ) = r ) ) -> ( E. z e. ( 1st ` A ) E. y e. ( 2nd ` A ) y <Q ( z +Q w ) -> ( q e. ( 1st ` C ) \/ r e. ( 2nd ` C ) ) ) )
98	10, 97	mpd 13	. . . . 5 |- ( ( ( A <P B /\ q <Q r ) /\ ( w e. Q. /\ ( q +Q w ) = r ) ) -> ( q e. ( 1st ` C ) \/ r e. ( 2nd ` C ) ) )
99	2, 98	rexlimddv 2628	. . . 4 |- ( ( A <P B /\ q <Q r ) -> ( q e. ( 1st ` C ) \/ r e. ( 2nd ` C ) ) )
100	99	ex 115	. . 3 |- ( A <P B -> ( q <Q r -> ( q e. ( 1st ` C ) \/ r e. ( 2nd ` C ) ) ) )
101	100	ralrimivw 2580	. 2 |- ( A <P B -> A. r e. Q. ( q <Q r -> ( q e. ( 1st ` C ) \/ r e. ( 2nd ` C ) ) ) )
102	101	ralrimivw 2580	1 |- ( A <P B -> A. q e. Q. A. r e. Q. ( q <Q r -> ( q e. ( 1st ` C ) \/ r e. ( 2nd ` C ) ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 710   /\ w3a 981   = wceq 1373   E.wex 1515   e. wcel 2176   A.wral 2484   E.wrex 2485   {crab 2488   <.cop 3636   class class class wbr 4045   ` cfv 5272  (class class class)co 5946   1stc1st 6226   2ndc2nd 6227   Q.cnq 7395   +Q cplq 7397   <Q cltq 7400   P.cnp 7406   <P cltp 7410

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-coll 4160  ax-sep 4163  ax-nul 4171  ax-pow 4219  ax-pr 4254  ax-un 4481  ax-setind 4586  ax-iinf 4637

This theorem depends on definitions:  df-bi 117  df-dc 837  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ne 2377  df-ral 2489  df-rex 2490  df-reu 2491  df-rab 2493  df-v 2774  df-sbc 2999  df-csb 3094  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3461  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-int 3886  df-iun 3929  df-br 4046  df-opab 4107  df-mpt 4108  df-tr 4144  df-eprel 4337  df-id 4341  df-po 4344  df-iso 4345  df-iord 4414  df-on 4416  df-suc 4419  df-iom 4640  df-xp 4682  df-rel 4683  df-cnv 4684  df-co 4685  df-dm 4686  df-rn 4687  df-res 4688  df-ima 4689  df-iota 5233  df-fun 5274  df-fn 5275  df-f 5276  df-f1 5277  df-fo 5278  df-f1o 5279  df-fv 5280  df-ov 5949  df-oprab 5950  df-mpo 5951  df-1st 6228  df-2nd 6229  df-recs 6393  df-irdg 6458  df-1o 6504  df-2o 6505  df-oadd 6508  df-omul 6509  df-er 6622  df-ec 6624  df-qs 6628  df-ni 7419  df-pli 7420  df-mi 7421  df-lti 7422  df-plpq 7459  df-mpq 7460  df-enq 7462  df-nqqs 7463  df-plqqs 7464  df-mqqs 7465  df-1nqqs 7466  df-rq 7467  df-ltnqqs 7468  df-enq0 7539  df-nq0 7540  df-0nq0 7541  df-plq0 7542  df-mq0 7543  df-inp 7581  df-iltp 7585

This theorem is referenced by:  ltexprlempr  7723